Let consider (a_n )_n and (u_n )_n two reals sequence defined such as a_0 =1 , ∀ n>1 a_(n+1) =Σ_(p=0) ^n a_p a_(n−p) and Σ_(p=0) ^n a_p u_(n−p) =0 Part1 1)Express ∀ n >1 a_n in terms of n 2) Find the largest domain of convergence of the integer serie {a_n x^n } 3)Determinate ∀ x∈D the sum f(x) of {a_n x^n } 4)Find the radius of convergence of the serie {u_n x^n } 5) Give the relation that between the sum S(x) of the second serie and (x/(f(x))) 6) Can you developp in integer serie g(x)=((πx)/(tan(πx))) Part2 Now do the part 1 but in the order 2)−1)−3)−4)−5)−6)


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Question Number 68220 by ~ À ® @ 237 ~ last updated on 07/Sep/19
$Let consider (a_n )_n and (u_n )_n two reals sequence defined such as a_0 =1 , ∀ n>1 a_(n+1) =Σ_(p=0) ^n a_p a_(n−p) and Σ_(p=0) ^n a_p u_(n−p) =0 Part1 1)Express ∀ n >1 a_n in terms of n 2) Find the largest domain of convergence of the integer serie {a_n x^n } 3)Determinate ∀ x∈D the sum f(x) of {a_n x^n } 4)Find the radius of convergence of the serie {u_n x^n } 5) Give the relation that between the sum S(x) of the second serie and (x/(f(x))) 6) Can you developp in integer serie g(x)=((πx)/(tan(πx))) Part2 Now do the part 1 but in the order 2)−1)−3)−4)−5)−6)$
$L e t c o n s i d e r {(a_{n})}_{n} a n d {(u_{n})}_{n} t w o r e a l s s e q u e n c e$ $d e f i n e d s u c h a s a_{0} = 1, \forall n > 1 a_{n + 1} = \underset{p = 0}{\sum^{n}} a_{p} a_{n - p} a n d \underset{p = 0}{\sum^{n}} a_{p} u_{n - p} = 0$ $P a r t 1$ $1) E x p r e s s \forall n > 1 a_{n} i n t e r m s o f n$ $2) F i n d t h e l a r g e s t d o m a i n o f c o n v e r g e n c e o f t h e i n t e g e r s e r i e {a_{n} x^{n}}$ $3) D e t e r m i n a t e \forall x \in D t h e s u m f (x) o f {a_{n} x^{n}}$ $4) F i n d t h e r a d i u s o f c o n v e r g e n c e o f t h e s e r i e {u_{n} x^{n}}$ $5) G i v e t h e r e l a t i o n t h a t b e t w e e n t h e s u m S (x) o f t h e s e c o n d s e r i e a n d \frac{x}{f (x)}$ $6) C a n y o u d e v e l o p p i n i n t e g e r s e r i e g (x) = \frac{π x}{t a n (π x)}$ $P a r t 2$ $N o w d o t h e p a r t 1 b u t i n t h e o r d e r 2) - 1) - 3) - 4) - 5) - 6)$
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